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| ==Generalized Eyring Relationship==
| | #REDIRECT [[Eyring_Relationship#Generalized_Eyring_Relationship]] |
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| ===Introduction===
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| <br>
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| The generalized Eyring relationship is used when temperature and a second non-thermal stress (e.g. voltage) are the accelerated stresses of a test and their interaction is also of interest. This relationship is given by:
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| <br>
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| ::<math>L(V,U)=\frac{1}{V}{{e}^{A+\tfrac{B}{V}+CU+D\tfrac{U}{V}}}</math>
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| <br>
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| :where:
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| <br>
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| • is the temperature (in absolute units ).
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| • <math>U</math> is the non-thermal stress (i.e. voltage, vibration, etc.).
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| <math>A,B,C,D</math> are the parameters to be determined.
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|
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| <br>
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| The Eyring relationship is a simple case of the generalized Eyring relationship where <math>C=D=0</math> and <math>{{A}_{Eyr}}=-{{A}_{GEyr}}.</math>
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| Note that the generalized Eyring relationship includes the interaction term of <math>U</math> and <math>V</math> as described by the <math>D\tfrac{U}{V}</math> term. In other words, this model can estimate the effect of changing one of the factors depending on the level of the other factor.
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| {{gen-eyring acceleration factor}}
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| {{generalized eyring-exponential}}
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| ===Generalized Eyring-Weibull===
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| <br>
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| By setting <math>\eta =L(V,U)</math> from Eqn. (Gen-Eyr), the generalized Eyring Weibull model is given by:
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| <br>
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| ::<math>\begin{align}
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| & f(t,V,U)= & \beta \left( V{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right){{\left( tV{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right)}^{\beta -1}} \\
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| & & .{{e}^{-{{\left( tV{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right)}^{\beta }}}}
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| \end{align}</math>
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| ====Generalized Eyring-Weibull Reliability Function====
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| The generalized Eyring Weibull reliability function is given by:
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| <br>
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| ::<math>R(T,V,U)={{e}^{-{{\left( tV{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right)}^{\beta }}}}</math>
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| ====Parameter Estimation====
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| Substituting the generalized Eyring model into the Weibull log-likelihood equation yields:
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| <br>
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| ::<math>\begin{align}
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| & \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [\beta \left( V{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right) \\
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| & & {{\left( tV{{e}^{-A-\tfrac{B}{V}-CU-D\tfrac{U}{V}}} \right)}^{\beta -1}}] \\
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| & & -\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{\left( {{t}_{i}}{{V}_{i}}{{e}^{-A-\tfrac{B}{{{V}_{i}}}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{{{V}_{i}}}}} \right)}^{\beta }} \\
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| & & -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }{{\left( t_{i}^{\prime }V_{i}^{\prime }{{e}^{-A-\tfrac{B}{V_{i}^{\prime }}-CU_{i}^{\prime }-D\tfrac{U_{i}^{\prime }}{V_{i}^{\prime }}}} \right)}^{\beta }} \\
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| & & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
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| \end{align}</math>
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| <br>
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| :where:
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| <br>
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| <br>
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| ::<math>R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime })={{e}^{-{{\left( T_{Li}^{\prime \prime }V_{i}^{\prime \prime }{{e}^{-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}}} \right)}^{\beta }}}}</math>
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| <br>
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| ::<math>R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime })={{e}^{-{{\left( T_{Ri}^{\prime \prime }V_{i}^{\prime \prime }{{e}^{-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}}} \right)}^{\beta }}}}</math>
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| <br>
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| :and:
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| <br>
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| • <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.
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| • <math>{{N}_{i}}</math> is the number of times-to-failure data points in the <math>{{i}^{th}}</math> time-to-failure data group.
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| • <math>A,B,C,D</math> are parameters to be estimated.
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| • <math>{{V}_{i}}</math> is the temperature level of the <math>{{i}^{th}}</math> group.
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| • <math>{{U}_{i}}</math> is the non-thermal stress level of the <math>{{i}^{th}}</math> group.
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| • <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.
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| • <math>S</math> is the number of groups of suspension data points.
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| • <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.
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| • <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.
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| • <math>FI</math> is the number of interval data groups.
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| • <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals.
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| • <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.
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| • <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.
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| <br>
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| <br>
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| The solution (parameter estimates) will be found by solving for the parameters <math>A,</math> <math>B,</math> <math>C,</math> and <math>D</math> so that <math>\tfrac{\partial \Lambda }{\partial A}=0,</math> <math>\tfrac{\partial \Lambda }{\partial B}=0,</math> <math>\tfrac{\partial \Lambda }{\partial D}=0</math> and <math>\tfrac{\partial \Lambda }{\partial D}=0</math> .
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| <br>
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| ===Generalized Eyring-Lognormal===
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| <br>
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| By setting <math>\sigma _{T}^{\prime }=L(V,U)</math> from Eqn. (Gen-Eyr), the generalized Erying lognormal model is given by:
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| <br>
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| ::<math>f(t,V,U)=\frac{\varphi (z(t))}{\sigma _{T}^{\prime }t}</math>
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| <br>
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| :where:
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| <br>
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| ::<math></math>
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| ====Generalized Eyring-Lognormal Reliability Function====
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| <br>
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| The generalized Erying lognormal reliability function is given by:
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| <br>
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| ::<math>R(T,V,U)=1-\Phi (z)</math>
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| ====Parameter Estimation====
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| <br>
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| Substituting the generalized Eyring model into the lognormal log-likelihood equation yields:
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| <br>
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| ::<math>\begin{align}
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| & \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [\frac{\varphi (z(t))}{\sigma _{T}^{\prime }t}]\overset{S}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\ln \left( 1-\Phi (z(t_{i}^{\prime })) \right) \\
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| & & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]
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| \end{align}</math>
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| <br>
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| :where:
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| <br>
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| ::<math>z_{Ri}^{\prime \prime }=\frac{\ln t_{Ri}^{\prime \prime }-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}+\ln (V_{i}^{\prime \prime })}{\sigma _{T}^{\prime }}</math>
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| <br>
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| ::<math>z_{Li}^{\prime \prime }=\frac{\ln t_{Ri}^{\prime \prime }-A-\tfrac{B}{V_{i}^{\prime \prime }}-C{{U}_{i}}-D\tfrac{{{U}_{i}}}{V_{i}^{\prime \prime }}+\ln (V_{i}^{\prime \prime })}{\sigma _{T}^{\prime }}</math>
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| <br>
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| :and:
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| <br>
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| • <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.
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| <br>
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| • <math>{{N}_{i}}</math> is the number of times-to-failure data points in the <math>{{i}^{th}}</math> time-to-failure data group.
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| <br>
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| • <math>A,B,C,D</math> are parameters to be estimated.
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| <br>
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| • <math>{{V}_{i}}</math> is the temperature level of the <math>{{i}^{th}}</math> group.
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| <br>
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| • <math>{{U}_{i}}</math> is the non-thermal stress level of the <math>{{i}^{th}}</math> group.
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| <br>
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| • <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.
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| <br>
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| • <math>S</math> is the number of groups of suspension data points.
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| <br>
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| • <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.
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| • <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.
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| • <math>FI</math> is the number of interval data groups.
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| <br>
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| • <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals.
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| • <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.
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| • <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.
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| <br>
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| <br>
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| The solution (parameter estimates) will be found by solving for the parameters <math>A,</math> <math>B,</math> <math>C,</math> and <math>D</math> so that <math>\tfrac{\partial \Lambda }{\partial A}=0,</math> <math>\tfrac{\partial \Lambda }{\partial B}=0,</math> <math>\tfrac{\partial \Lambda }{\partial D}=0</math> and <math>\tfrac{\partial \Lambda }{\partial D}=0</math> .
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| <br>
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| ===Example===
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| <br>
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| The following data set represents failure times (in hours) obtained from an electronics epoxy packaging accelerated life test performed to understand the synergy between temperature and humidity and estimate the <math>B10</math> life at the use conditions of <math>T=350K</math> and <math>H=0.3</math> . The data set is modeled using the lognormal distribution and the generalized Eyring model.
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| <br>
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| [[Image:altaproject.png|thumb|center|400px| ]]
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| [[Image:altaaccelerated.png|thumb|center|400px| ]]
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| [[Image:altatimes2fail.png|thumb|center|400px|]]
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| [[Image:tempNhumin.png|thumb|center|400px|]]
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| [[Image:406-1-2.png|thumb|center|400px|]]
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| <math></math>
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| The probability plot at the use conditions is shown next.
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| <br>
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| [[Image:plotfolio426.png|thumb|center|400px|]]
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| <br>
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|
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| The <math>B10</math> information is estimated to be 3004.63 hours, as shown next.
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| <br>
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| [[Image:tempBX.png|thumb|center|400px|]] | |
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| <br>
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